# Is the elementary transformation of a conic bundle a flip or a flop

Let $$\pi: V\to S$$ be a standard conic bundle of a threefold $$V$$ to a surface $$S$$, i.e., $$\pi$$ is relative minimal. Assume that everything is nonsingular and is over $$\mathbb{C}$$. We may assume that $$V$$ is embedded in a $$\mathbb{P}^2$$-bundle $$\mathbb{P}(\mathcal{E})$$ over $$S$$, where $$\mathcal{E}$$ is a rank $$3$$ vector bundle on $$S$$, and $$V$$ is defined as a zero of a section $$\sigma\in H^0(\mathbb{P}(E),\mathcal{O}_{\mathbb{P}(E)}(2)\otimes\tau^*\mathcal{L})$$, where $$\mathcal{L}$$ is an invertible sheaf on $$S$$ and $$\tau$$ is the standard projection of $$\mathbb{P}(\mathcal{E})$$ to $$S$$.

Now suppose that $$C$$ is a curve on $$V$$ which is isomorphic to its image $$\pi(C)$$, such that $$\pi(C)\cap\Delta=\varnothing$$, where $$\Delta$$ is the degenerate divisor on $$S$$ for the conic bundle $$(V, S,\pi)$$, that is the locus of points whose fiber is a degenerate conic, disjoint of two lines or a double line. Then we can apply an elementary transformation to $$V$$, as described in this paper On conic bundle structures--V. G. Sarkisov, which first blows up $$C$$ then blows down the strict transform of $$B=\pi^{-1}(\pi(C))$$. The resulting variety is denoted by $$V'$$, and $$C'$$ is the strict transform of $$C$$.

Question: Is this elementary transformation a flop or a flip in the sense of the MMP?

I try to answer this question positively, in other words, does the intersection numbers hold $$K_V.C<0$$ and $$K_{V'}.C'>0$$? But I meet some trouble when I verify this inequality, by 1.5 of On conic bundle structures--V. G. Sarkisov, we know that $$\mathcal{L}$$ is isomorphic to $$\omega_S\otimes\mathrm{det}(\mathcal{E})$$, so $$\omega_{\mathbb{P}(\mathcal{E})}\otimes\mathcal{O}_{\mathbb{P}(\mathcal{E})}(V)\cong\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)$$. Therefore the intersection number of $$K_V.C$$ should be equal to $$(K_{\mathbb{P}(\mathcal{E})}+V).C=\mathrm{deg}_C\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)$$, as $$C$$ contained in $$V$$. We only konw that the divisor associate to $$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)$$ is relatively ample, but here the curve $$C$$ is not in a fiber, we can not conclude that $$\mathrm{deg}_C\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)<0$$ directly.

On the other hand, it seems that the situation for $$(V',S,\pi')$$ is the same as $$(V,S,\pi)$$. Since $$(V',S,\pi')$$ is also standard. So we have $$K_{V'}.C'=\mathrm{deg}_{C'}\mathcal{O}_{\mathbb{P}(\mathcal{E})}(-1)$$. Note that $$B$$ (resp. $$B'=\pi'^{-1}(\pi'(C'))$$) is a ruled surface over $$\pi(C)$$ (resp. $$\pi'(C)$$), if my question has a positive anwser, it seems that $$C$$ is the negative section of $$B\to\pi(C)$$ and $$C'$$ is a positive section of $$B'\to \pi'(C')$$, i.e., $$C^2<0$$ and $$C'^2>0$$. Let $$W\to V$$ be the blowing up along $$C$$, and let $$\tilde{B}$$ and $$\tilde{B'}$$ be the strict transforms of $$B$$ and $$B'$$ respectively. It seems that $$\tilde{B}$$ and $$\tilde{B'}$$ are guling along a positive section of $$\tilde{B}$$ with a negative section of $$\tilde{B'}$$ in $$W$$. But I don't know how to prove this. However, it may be false.

• It is not a flip or flop (since there is no extremal contraction of the curve $C\subset V$ in this setting). Instead it is an example of a Sarkisov link (see, eg., Hacon & McKernan, `The Sakisov Program'). Apr 12, 2021 at 10:57

It is neither flip nor flop, because the exceptional locus both on $$V$$ and on $$V'$$ is a divisor.